/* ==================================== */
/* Linear Least Square Fit to Data in Table 6.1 of Bevington and Robinson.

   The data is the voltage vs position along a wire, and is fit to a model
   of V[i] = a*x[i] + b
 */

/* In math sum is a linear function, but maxima is very conservative.  By
 * default, it assumes sum *might* be non-linear */
declare(sum,linear);

/* Set the data */
n : 9;
V : [0.37, 0.58, 0.83, 1.15, 1.36, 1.62, 1.90, 2.18, 2.45];
x : [10.0, 20.0, 30.0, 40.0, 50.0, 60.0, 70.0, 80.0, 90.0];
%sigma : [0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05];

/* Define a chi2 to find the general solution. */
chi2: sum((V[i] - a*x[i] - b)^2/%sigma[i]^2, i, 1, n);

/* Find the derivatives wrt a and b */
Dchi2Da : expand(diff(chi2,a));
Dchi2Db : expand(diff(chi2,b));


/* Invert the error matrix (or inverse covariance) to find the covariance. */
cov : invert(matrix( [diff(Dchi2Da,a), diff(Dchi2Da,b)],
	                [diff(Dchi2Db,a), diff(Dchi2Db,b)]) / 2);

/* Solve for the parameters at the minimum chi2 value*/
soln: float(solve([Dchi2Da=0, Dchi2Db=0], [a,b]));

/* Calculate the minimum chi2 value. */
float(ev(chi2, soln[1][1], soln[1][2]));